標本分散の期待値

$$\begin{array}{rcl}{s}^2& =& {\displaystyle \frac{1}{n}\sum _{k=1}^n{\left({\displaystyle X_k-\bar{X}{\displaystyle }}\right)}^2\dots \mathrm{標}\mathrm{本}\mathrm{分}\mathrm{散}(samplevariance)}\\ E\left[s^2\right]& =& E\left[{\displaystyle \frac{1}{n}\sum _{k=1}^n{(X_k-\bar{X})}^2}\right]\\ & =& {\displaystyle \frac{1}{n}E\left[\sum _{k=1}^n{(X_k-\bar{X})}^2\right]}\\ & =& {\displaystyle \frac{1}{n}(n-1){\sigma }^2\dots {\displaystyle E\left[\sum _{k=1}^n{(X_k-\bar{X})}^2\right]=(n-1){\sigma }^2}}\\ & =& {\displaystyle \frac{n-1}{n}{\sigma }^2}\end{array}$$